Combinatorics: the Art of Counting

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GRADUATE STUDIES IN MATHEMATICS 210

Combinatorics: The Art of Counting

Bruce E. Sagan

Combinatorics: The Art of Counting

GRADUATE STUDIES IN MATHEMATICS 210

Combinatorics: The Art of Counting

Bruce E. Sagan Marco Gualtieri Bjorn Poonen Gigliola Staﬃlani (Chair) Jeﬀ A. Viaclovsky

2020 Mathematics Subject Classiﬁcation. Primary 05-01; Secondary 06-01.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-210

Library of Congress Cataloging-in-Publication Data Names: Sagan, Bruce Eli, 1954- author. Title: Combinatorics : the art of counting / Bruce E. Sagan. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Gradu- ate studies in mathematics, 1065-7339 ; 210 | Includes bibliographical references and index. Identiﬁers: LCCN 2020025345 | ISBN 9781470460327 (paperback) | ISBN 9781470462802 (ebook) Subjects: LCSH: Combinatorial analysis–Textbooks. | AMS: Combinatorics – Instructional exposition (textbooks, tutorial papers, etc.). | Order, lattices, ordered algebraic structures – Instructional exposition (textbooks, tutorial papers, etc.). Classiﬁcation: LCC QA164 .S24 2020 | DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2020025345

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Contents

Preface xi

List of Notation xiii

Chapter 1. Basic Counting 1 §1.1. The Sum and Product Rules for sets 1 §1.2. Permutations and words 4 §1.3. Combinations and subsets 5 §1.4. Set partitions 10 §1.5. Permutations by cycle structure 11 §1.6. Integer partitions 13 §1.7. Compositions 16 §1.8. The twelvefold way 17 §1.9. Graphs and digraphs 18 §1.10. Trees 22 §1.11. Lattice paths 25 §1.12. Pattern avoidance 28 Exercises 33

Chapter 2. Counting with Signs 41 §2.1. The Principle of Inclusion and Exclusion 41 §2.2. Sign-reversing involutions 44 §2.3. The Garsia–Milne Involution Principle 49 §2.4. The Reflection Principle 52

vii viii Contents

§2.5. The Lindström–Gessel–Viennot Lemma 55 §2.6. The Matrix-Tree Theorem 59 Exercises 64

Chapter 3. Counting with Ordinary Generating Functions 71 §3.1. Generating polynomials 71 §3.2. Statistics and 푞-analogues 74 §3.3. The algebra of formal power series 81 §3.4. The Sum and Product Rules for ogfs 86 §3.5. Revisiting integer partitions 89 §3.6. Recurrence relations and generating functions 92 §3.7. Rational generating functions and linear recursions 96 §3.8. Chromatic polynomials 99 §3.9. Combinatorial reciprocity 106 Exercises 109

Chapter 4. Counting with Exponential Generating Functions 117 §4.1. First examples 117 §4.2. Generating functions for Eulerian polynomials 121 §4.3. Labeled structures 124 §4.4. The Sum and Product Rules for egfs 128 §4.5. The Exponential Formula 131 Exercises 134

Chapter 5. Counting with Partially Ordered Sets 139 §5.1. Basic properties of partially ordered sets 139 §5.2. Chains, antichains, and operations on posets 145 §5.3. Lattices 148 §5.4. The Möbius function of a poset 154 §5.5. The Möbius Inversion Theorem 157 §5.6. Characteristic polynomials 164 §5.7. Quotients of posets 168 §5.8. Computing the Möbius function 174 §5.9. Binomial posets 178 Exercises 183

Chapter 6. Counting with Group Actions 189 §6.1. Groups acting on sets 189 §6.2. Burnside’s Lemma 192 §6.3. The cycle index 197 Contents ix

§6.4. Redfield–Pólya theory 200 §6.5. An application to proving congruences 205 §6.6. The cyclic sieving phenomenon 209 Exercises 213

Chapter 7. Counting with Symmetric Functions 219 §7.1. The algebra of symmetric functions, Sym 219 §7.2. The Schur basis of Sym 224 §7.3. Hooklengths 230 §7.4. 푃-partitions 235 §7.5. The Robinson–Schensted–Knuth correspondence 240 §7.6. Longest increasing and decreasing subsequences 244 §7.7. Differential posets 248 §7.8. The chromatic symmetric function 253 §7.9. Cyclic sieving redux 256 Exercises 259

Chapter 8. Counting with Quasisymmetric Functions 267 §8.1. The algebra of quasisymmetric functions, QSym 267 §8.2. Reverse 푃-partitions 270 §8.3. Chain enumeration in posets 274 §8.4. Pattern avoidance and quasisymmetric functions 276 §8.5. The chromatic quasisymmetric function 279 Exercises 283

Appendix. Introduction to Representation Theory 287 §A.1. Basic notions 287 Exercises 292

Bibliography 293

Index 297

Preface

Enumerative combinatorics has seen an explosive growth over the last 50 years. The purpose of this text is to give a gentle introduction to this exciting area of research. So, rather than trying to cover many different topics, I have chosen to give a more leisurely treatment of some of the highlights of the field. My goal has been to write the exposition so it could be read by a student at the advanced undergraduate or beginning graduate level, either as part of a course or for independent study. The reader will find it similar in tone to my book on the symmetric group. I have tried to keep the prerequisites to a minimum, assuming only basic courses in linear and abstract algebra as background. Certain recurring themes are emphasized, for example, the existence of sum and product rules first for sets, then for ordinary generating functions, and finally inthecaseof exponential generating functions. I have also included some recent material from the research literature which, to my knowledge, has not appeared in book form previously, such as the theory of quotient posets and the connection between pattern avoidance and quasisymmetric functions. Most of the exercises should be doable with a reasonable amount of effort. A few unsolved conjectures have been included among the problems in the hope that an interested student might wish to tackle one of them. They are, of course, marked as such. A few words about the title are in order. It is in part meant to be a tip of the hat to Donald Knuth’s influential series of books The art of computer programing, Volumes 1–3 [51–53], which, among many other things, helped give birth to the study of pattern avoidance through its connection with stack sorting; see Exercise 36 in Chapter 1. I hope that the title also conveys some of the beauty found in this area of mathematics, for example, the elegance of the Hook Formula (equation (7.10)) for the number of standard Young tableaux. In addition I should mention that, due to my own pref- erences, this book concentrates on the enumerative side of combinatorics and mostly ignores the important extremal and existential parts of the field. The reader interested in these areas can consult the books of Flajolet and Sedgewick [25] and of van Lint [95].

xi xii Preface

This book grew out of the lecture notes which I have compiled over years of teaching the graduate combinatorics course at Michigan State University. I would like to thank the students in these classes for all the feedback they have given me about the various topics and their presentation. I am also indebted to the following colleagues, some of whom taught from a preliminary version of this book, who provided me with suggestions as well as catching numerous typographical errors: Matthias Beck, Moussa Benoumhani, Andreas Blass, Seth Chaiken, Sylvie Corteel, Georges Grekos, Richard Hensh, Nadia Lafrenière, Duncan Levear, and Tom Zaslavsky. Darij Grinberg deserves special mention for providing copious comments and corrections as well as providing a number of interesting exercises. I also received valuable feedback from four anony- mous referees. Finally, I wish to express my appreciation of Ina Mette, my editor at the American Mathematical Society. Without her gentle support and persistence, this text would never have seen the light of day. Because I typeset this document myself, all errors can be blamed on my computer.

East Lansing, Michigan, 2020 List of Notation

Symbol Definition Page

퐴(퐷) arc set of digraph 퐷 21 퐴(퐺) adjacency matrix of graph 퐺 60 풜(퐺) set of acyclic orientations of 퐺 103 푎(퐺) number of acyclic orientations of 퐺 103 퐴([푛], 푘) set of permutations 휋 in 픖푛 having 푘 descents 121 퐴(푛, 푘) Eulerian number, cardinality of 퐴([푛], 푘) 121 퐴푛(푞) Eulerian polynomial 122 풜(푃) atom set of poset 푃 169 Asc 푐 ascent set of a proper coloring 푐 279 asc 푐 ascent number of a proper coloring 푐 279 Asc 휋 ascent set of permutation 휋 76 asc 휋 ascent number of permutation 휋 76 Av푛(휋) the set of permutations in 픖푛 avoiding 휋 29 훼푟 reversal of composition 훼 32 ̄훼 expansion of composition 훼 274 훼(퐶) rank composition of chain 퐶 275 퐵(퐺) incidence matrix of graph 퐺 61 퐵(푇) set of partitions of the set 푇 10 퐵푛 Boolean algebra on [푛] 140 퐵∞ poset of subsets of ℙ 178 퐵(푛) 푛th Bell number 10

xiii xiv List of Notation

Symbol Definition Page

ℂ complex numbers 1 푐푖(푔) number of cycles of length 푖 in group element 푔 197 퐶퐿푛 claw poset with 푛 atoms 169 co 푇 content of tableau 푇 225 퐶푛 cycle with 푛 vertices 19 퐶푛 chain poset of length 푛 139 푐푥(푃) column insertion of element 푥 into tableau 푃 245 퐶∞ chain poset on ℕ 178 퐶(푛) Catalan number 26 푐([푛], 푘) set of permutations in 픖푛 with 푘 cycles 12 푐(푛, 푘) signless Stirling number of the first kind 12 푐표(퐿, 푘) ordered 푘 cycle decompositions of permutations of 퐿 127 ℂ푋 vector space generated by set 푋 over ℂ 248 ℂ[푥] polynomial algebra in 푥 over ℂ 71 ℂ[[푥]] formal power series algebra in 푥 over ℂ 81 풞(휋) set of functions compatible with 휋 236 풞푚(휋) set of functions compatible with 휋 bounded by 푚 236 Des 푃 descent set of tableau 푃 271 Des 휋 descent set of permutation 휋 75 des 휋 descent number of permutation 휋 76 퐷푛 lattice of divisors of 푛 140 퐷∞ divisibility poset on ℙ 181 퐷(푛) derangement number 43 풟(푛) set of Dyck paths of semilength 푛 26 풟(푉) set of all digraphs on vertex set 푉 21 풟(푉, 푘) set of all digraphs on vertex set 푉 with 푘 edges 21 deg 푚 degree of a monomial 219 deg 푣 degree of vertex 푣 in a graph 20 Δ푓(푛) forward difference operator of 푓(푛) 162 훿푥,푦 Kronecker delta 7 훿(푥, 푧) delta function of poset incidence algebra 159 퐸(퐺) edge set of graph 퐺 18 퐸(퐿) set structure on label set 퐿 125 퐸(퐿) nonempty set structure on label set 퐿 125 퐸푛 Euler number 120 푒푛 푛th elementary symmetric function 221 퐸(푡) generating function for elementary symmetric functions 221 Exc 휋 set of excedances of permutation 휋 122 exc 휋 number of excedances of permutation 휋 122 List of Notation xv

Symbol Definition Page

Fix 푓 fix point set of a function 푓 44 푓푛 Fibonacci number 3 퐹푛 Fibonacci number 2 픽푞 Galois field with 푞 elements 79 푓(푥) ordinary generating function 81 푓푆(푥) weight-generating function for weighted set 푆 86 퐹(푛) binomial poset 푛-interval factorial function 178 퐹(푥) exponential generating function 117 퐹풮(푥) exponential generating function for structure 풮 125 퐹푆 fundamental quasisymmetric for set 푆 269 퐹훼 fundamental quasisymmetric for composition 훼 269 푓휆 number of standard Young tableaux of shape 휆 225 Φ fundamental map on permutations 122 휙 bijection between subsets and compositions 16 퐺 ⧵ 푒 graph 퐺 with edge 푒 deleted 100 퐺/푒 graph 퐺 with edge 푒 contracted 101 GL(푉) general linear group over vector space 푉 287 풢(푉) set of all graphs on vertex set 푉 20 풢(푉, 푘) set of all graphs on vertex set 푉 with 푘 edges 20 퐺푥 stabilizer of element 푥 under the action of group 퐺 191 퐻푐 = 퐻푖,푗 hook of cell 푐 = (푖, 푗) 230 ℎ푐 = ℎ푖,푗 hooklength of cell 푐 = (푖, 푗) 230 ℋ푛 set of hook diagrams with 푛 cells 278 ℎ푛 푛th complete homogeneous symmetric function 221 퐻(푡) complete homogeneous generating function 221 ideg 푣 in-degree of vertex 푣 in a digraph 21 Inv 휋 inversion set of permutation 휋 74 inv 휋 inversion number of permutation 휋 74 ℐ(푃) incidence algebra of poset 푃 158 퐼(푆) lower-order ideal generated by 푆 in a poset 143 ISF(퐺; 푡) increasing spanning forest generating function of 퐺 105 ISF푚(퐺) set of 푚-edge increasing spanning forests of 퐺 105 isf푚(퐺) number of 푚-edge increasing spanning forests of 퐺 105 푖휆(퐺) number of independent type 휆 partitions in graph 퐺 254 풥(푃) distributive lattice of lower-order ideals of poset 푃 151 퐾푛 complete graph with 푛 vertices 19 퐾푛 lattice of compositions of 푛 140 퐾휆,휇 number of tableaux of shape 휆 and content 휇 225 xvi List of Notation

Symbol Definition Page

퐿(퐺) Laplacian of graph 퐺 62 ℒ(퐺) bond lattice of graph 퐺 167 ℒ(푃) set of linear extensions of 푃 238 ℓ(퐶) length of chain 퐶 in a poset 147 ℓ(휆) length of an integer partition 휆 15 ℓ(휋) length of a permutation 휋 4 lim 푓푘(푥) limit of a sequence of formal power series 84 푘→∞ lds 휋 length of a longest decreasing subsequence of 휋 245 lis 휋 length of a longest increasing subsequence of 휋 244 푛 퐿푛(푞) lattice of subspaces of 픽푞 140 퐿∞(푞) poset of subspaces of vector space 푉∞ over 픽푞 178 퐿(푉) lattice of subspaces of 푉 140 휆(퐹) type of partition induced by edge set 퐹 255 휆! multiplicity factorial of partition 휆 254 maj 휋 major index of permutation 휋 76 푀(푛) Mertens function 183 푀(푃) monomial quasisymmeric function for poset 푃 275 푀훼 monomial quasisymmetric function 268 푚휆 monomial symmetric function 220 휇(푃) Möbius function value on a poset 푃 154 휇(푥) one-variable Möbius function evaluated at 푥 154 휇(푥, 푧) two-variable Möbius function on the interval [푥, 푧] 157 ℕ nonnegative integers 1 NBC푘(퐺) set of no broken circuit sets of 푘 edges of 퐺 102 nbc푘(퐺) number of no broken circuit sets of 푘 edges of 퐺 102 풩ℰ(푚, 푛) set of 푁-퐸 lattice paths from (0, 0) to (푚, 푛) 26 odeg 푣 out-degree of vertex 푣 in a digraph 21 풪푥 orbit of an element 푥 under action of a group 190 푂(푔) big oh notation applied to function 푔 182 표(푔) order of a group element 푔 210 ℙ positive integers 1 푃∗ dual of poset 푃 142 풫퐶(퐺) set of proper colorings of 퐺 with the positive integers 279 푃(퐺; 푡) chromatic polynomial of graph 퐺 100 Par 푃 set of 푃-partitions 238 Par푚 푃 set of 푃-partitions bounded by 푚 238 푃푛 path with 푛 vertices 19 푃(푛) set of partitions of the integer 푛 13 푝(푛) number of partitions of the integer 푛 13 푝푛 푛th power sum symmetric function 221 푃(푡) power sum symmetric generating function 221 List of Notation xvii

Symbol Definition Page

푃(푛, 푘) set of partitions of 푛 into at most 푘 parts 15 푝(푛, 푘) number of partitions of 푛 into at most 푘 parts 15 푃(푆) permutations of a set 푆 4 푃(푆, 푘) permutations of length 푘 of a set 푆 4 푃((푆, 푘)) words of length 푘 over a set 푆 5 푃(휋) insertion tableau of 휋 242 풫(푢; 푣) set of directed paths from 푢 to 푣 in a digraph 56 Π푛 partition lattice on [푛] 140 Π(풮) partition structure on structure 풮 131 Π푒(풮) even partition structure on structure 풮 133 Π표(풮) odd partition structure on structure 풮 133 ℚ rational numbers 1 푄(푛) set of compositions of the integer 푛 16 푞(푛) number of compositions of the integer 푛 16 푄(푛, 푘) set of compositions of 푛 into 푘 parts 16 푞(푛, 푘) number of partitions of 푛 into 푘 parts 16 QSym algebra of quasisymmetric functions 268

QSym푛 quasisymmetric functions of degree 푛 268 푄(휋) recording tableau of 휋 242 푄푛(Π) quasisymmetric function for patterns Π 277 ℝ real numbers 1 ℛ퐶(휋) set of functions reverse compatible with 휋 270 rk 푃 rank of a ranked poset 푃 147 Rk푘 푃 푘th rank set of a ranked poset 푃 147 rk 푥 rank of an element 푥 in a ranked poset 147 ℛ(푘, 푙) set of partitions contained in a 푘 × 푙 rectangle 79 RPar 푃 set of reverse 푃-partitions 271 ℛ(푃) reduced incidence algebra of a binomial poset 179

rpp푛(휆) number of shape 휆 reverse plane partitions of 푛 233 rpar(푃; 퐱) generating function for reverse 푃-partitions 271 푟푥(푃) row insertion of element 푥 into tableau 푃 241 휌(퐹) vertex partition induced by edge set 퐹 255 휌∶ 퐺 → GL(푉) representation of group 퐺 287 풮(퐿) labeled structure on label set 퐿 124 픖 pattern poset 140 픖푛 symmetric group on [푛] 11 푆푓(푛) summation operator applied to function 푓(푛) 162 sgn sign function on a signed set 44 sh 푇 shape of tableau 푇 225 푠(푛, 푘) signed Stirling number of the first kind 13 xviii List of Notation

Symbol Definition Page

푆(푇, 푘) set of partitions of the set 푇 into 푘 blocks 10 푆(푛, 푘) Stirling number of the second kind 10 푆표(퐿, 푘) set of ordered partitions of the set 퐿 into 푘 blocks 127 풮푇(퐺) set of spanning trees of graph 퐺 59 st statistic on a set 74 std 휎 standardization of the permutation 휎 28 Supp 푥 support set of 푥 in a product of claws 173 supp 푥 size of support set of 푥 in a product of claws 173 Sym algebra of symmetric functions 220

Sym푛 symmetric functions of degree 푛 220 SYT(휆) set of standard Young tableaux of shape 휆 224 SSYT(휆) set of semistandard Young tableaux of shape 휆 225 푠휆 Schur function 225 푇푖,푗 element in cell (푖, 푗) of tableau 푇 225 풯푛 set of monomino-domino tilings of a row of 푛 squares 3 푈(푆) upper-order ideal generated by 푆 in a poset 143 푉(퐷) vertex set of digraph 퐷 21 푉(퐺) vertex set of graph 퐺 18 푉∞ vector space with a countably infinite basis over 픽푞 178 푤푘(푃) Whitney number of the first kind for a poset 푃 156 푊 푘(푃) Whitney number of the second kind for a poset 푃 156 푊푛 walk with 푛 vertices 19 wt weight function on a set 86 퐱 a countably infinite set of variables 219 퐱푐 monomial for a coloring 푐 of a graph 253 퐱푓 monomial for a function 푓 270 퐱푇 monomial for a tableau 푇 225 푋푔 fixed points of group element 푔 acting on set 푋 192 푋(퐺; 퐱) chromatic symmetric function of graph 퐺 253 푋(퐺; 퐱, 푞) chromatic quasisymmetric function of graph 퐺 280 푌 Young’s lattice 140 ℤ set of integers 1 휁(푥, 푧) zeta function in the incidence algebra of a poset 159 휁(푠) Riemann zeta function 182 푧(푔) cycle index of group element 푔 197 푍(퐺) cycle index of group 퐺 197 #푆 cardinality of the set 푆 1 |푓| size (sum of values) of a function 236 |푆| cardinality of the set 푆 1 |푇| sum of entries of tableau 푇 233 푆 ⊎ 푇 disjoint union of sets 푆 and 푇 1 List of Notation xix

Symbol Definition Page

|휆| sum of the parts of partition 휆 13 휆 ⊢ 푛 휆 is a partition of 푛 13 푆 × 푇 (Cartesian) product of sets 푆 and 푇 1 푃 ⊎ 푄 disjoint union of posets 푃 and 푄 145 푃 ⊕ 푄 ordinal sum of posets 푃 and 푄 146 푃 × 푄 (Cartesian) product of posets 푃 and 푄 146 [푔] linear transformation for group element 푔 287 [푔]퐵 matrix in basis 퐵 for group element 푔 287 [푛] set of integers {1, 2, ... , 푛} 4 [푛]푞 푞-analogue of nonnegative integer 푛 75 [푛]푞! 푞-analogue of 푛! 75 [푥푛]푓(푥) coefficient of 푥푛 in 푓(푥) 83 푛↓푘 푛 falling factorial with 푘 factors 4 2푆 set of subsets of 푆 5 푆 (푘) set of 푘-element subsets of 푆 6 푛 (푘) binomial coefficient 7 [푛] 푞-binomial coefficient 77 푘 푞 푉 [푘] 푘-dimensional subspaces of vector space 푉 79 {{푎, 푎, ... }} multiset individual element notation 8 {{푎2, ... }} multiset multiplicity notation 8 푆 ((푘)) set of 푘-element multisubsets of 푆 9 휒(퐺) chromatic number of 퐺 99 휒(푔) character of group element 푔 291 푥 ⋖ 푦 푥 is covered by 푦 in a poset 140 푦 ⋗ 푥 푦 covers 푥 in a poset 140 0̂ the minimum element of a poset 142 1̂ the maximum element of a poset 142 [푥, 푦] closed interval from 푥 to 푦 in a poset 143 푥 ∧ 푦 meet of 푥 and 푦 in a poset 148 ⋀ 푋 meet of the subset 푋 in a poset 149 푥 ∨ 푦 join of 푥 and 푦 in a poset 149 푈 + 푉 sum of subspaces 푈 and 푉 149 푓 ∗ 푔 convolution of 푓 and 푔 in the incidence algebra 158 휒(푃; 푡) characteristic polynomial of a ranked poset 푃 164 푃/ ∼ quotient of poset 푃 by equivalence relation ∼ 169 휔푛 primitive 푛th root of unity 210 RS 휋 ↦ (푃, 푄) Robinson–Schensted map 242 RSK 푀 ↦ (푇, 푈) Robinson–Schensted–Knuth map 244

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푘-permutation, 4 poset, 178 푘-word, 5 Theorem, 72 푞-analogue, 75 푞-analogue, 78 binomial coefficient, 77 negative, 88 Binomial Theorem, 78 Birkhoff, George factorial, 75 distributive lattice, 151 natural number, 75 Björner, Anders topological methods, 176 al-Haytham, Ibn Block, 10 Wilson’s Congruence, 206 Bloom, Jonathan Algebra quasisymmetric pattern avoidance, 276 formal power series, 81 Broken circuit, 102 Algorithm Burnside, William Greene–Nijenhuis–Wilf, 231 Lemma, 192 Hillman–Grassl, 234 Burton Prüfer, 24 Fermat’s Little Theorem, 206 Robinson–Schensted, 241 Robinson–Schensted–Knuth, 244 Catalan number, 26 Appel, Kenneth Cauchy–Binet Theorem, 63 Four Color Theorem, 100 Coefficient Ascent, 76 binomial, 7 Avoid a permutation, 28 Coexcedance, 123 Composition, 16 Babson, Eric dominance, 31 Mahonian statistics, 76 expansion, 274 Baker-Jarvis, Duff part, 16 shuffle compatibility, 274 set, 47 Beck, Matthias merge, 48 combinatorial reciprocity, 107 split, 47 Bell number, 10 shuffle sum, 274 Bender, Edward weak, 31 Schur function, 226 Condition Big oh notation, 182 Rank, 171 Bijective proof, 6 Summation, 171 Binomial Congruence, 205 coefficient, 7 Fermat, 206 푞-analogue, 77 Lucas, 207

297 298 Index

Möbius function, 208 Fermat, Pierre Wilson, 207 Little Theorem, 206 Conjugate Ferrers diagram, 14 Young diagram, 14 Fibonacci number, 2 Copy of a permutation, 28 Fixed point, 11 Cycle Flajolet, Philippe decomposition, 11 analytic combinatorics, 81 digraph, 21 Foata, Dominique graph, 19 Mahonian statistic, 76 permutation, 11 major index, 76 Cyclic sieving phenomenon (CSP), 210, 256 Forest, 22 Formal power series, 81 Degree algebra, 81 graph vertex, 20 convergence Deletion-Contraction Lemma, 101 product, 85 Derangement, 43 sequence, 83 Descent, 75 sum, 84 Diagram degree, 219 Ferrers, 14 bounded, 219 Hasse, 140 homogeneous, 219 permutation, 29 minimum degree, 84 Young, see also Young, diagram quasisymmetric, 267 Digraph symmetric, 220 acyclic, 56 Formula arc, 21 exponential, 132 multiple, 22 Hook, 231 cycle, 21 Four Color Theorem, 100 functional, 22 Frame, J. Sutherland in-degree, 21 Hook Formula, 230 labeled, 21 Functional digraph, 22 loop, 22 orientation, 103 Garsia, Adriano out-degree, 21 Involution Principle, 49 path, 21 Garsia–Milne Involution Principle, 49 walk, 21 Gaussian polynomial, 77 Dihedral group, 29 Generating function Directed Graph, see also Digraph Eulerian, 178 Distribution, 74 exponential, 117 Doubilet, Peter method of undetermined coefficients, 99 binomial poset, 178 ordinary, 81 Dyck path of semilength 푛, 26 rational, 96 weight, 86 Ehrenborg, Richard Generating polynomial, 71 chains in posts, 274 Gessel, Ira Euler, Leonhard lattice paths, 58 Fermat’s Little Theorem, 206 quasisymmetric function, 267 number, 120 shuffle compatibility, 274 partition theorem, 51, 92 Graph Eulerian acyclic, 22 generating function, 178 bond, 167 number, 121 bond lattice, 167 polynomial, 122 broken circuit, 102 statistic, 121 chromatic Excedance, 122 number, 99 Exponential Formula, 132 polynomial, 100 quasisymmetric function, 280 Falling factorial, 4 symmetric function, 253 Index 299

clique, 19 cyclic sieving, 210, 256 coloring, 99 fixed point set, 192 complete, 19 induced component, 22 on functions, 193 connected, 22 on permutations, 197 cycle, 19 on subsets, 197 cycle of length ℓ, 19 orbit, 190 digraph, see also Digraph weight, 201 edge, 18 Redfield–Pólya Theorem, 201 multiple, 22 representation, see also Representation endpoints, 19 stabilizer, 191 forest, 22 weight increasing, 104 of an element, 200 Four Color Theorem, 100 of an orbit, 201 labeled, 18 leaf, 23 Haken, Wolfgang loop, 22 Four Color Theorem, 100 matrix Hall, Phillip adjacency, 60 Möbius function, 175 incidence, 61 Hallam, Joshua Laplacian, 62 increasing spanning forest, 105 no broken circuit (NBC), 102 quotient poset, 168, 174 orientation, 103 Hamaker, Zachary path, 19 quasisymmetric pattern avoidance, 276 proper coloring, 99 Handshaking Lemma, 21 ascent, 279 Hillman, Abraham subgraph, 19 reverse plane partition, 233 induced, 166 In-degree, 21 induced by a coloring, 167 Increasing forest, 104 spanning, 59 Integer partition, 13 tree, 22 containment, 78 trivial, 19 distinct parts, 51 unlabeled, 20 hook, 230 generating function, 202 hooklength, 230 vertex, 18 length, 15 degree, 20 lexicographic order, 222 walk of length ℓ, 19 multiplicity notation, 14 Grassl, Richard odd parts, 51 reverse plane partition, 233 part, 13 Greene, Curtis rectangule, 79 and Schensted’s Theorem, 248 skew, 78 Hook Formula, 231 Inversion, 74 Grinberg, Darij Involution, 44 shuffle compatibility, 274 sign reversing, 44 Group dihedral, 29 Jacobi, Carl general linear, 287 determinants, 227 order, 210 James, Gordon representation, see also Representation representation theory, 287 symmetric, 11 Joyal, André Group action, 189 species, 124 Burnside’s Lemma, 192 congruence, 205 Kerber, Adalbert cycle index representation theory, 287 of a group, 197 Knuth, Donald of an element, 197 Robinson–Schensted–Knuth algorithm, 244 cycle indicator, 197 Schur function, 226 300 Index

subspace, 81 on a set, 9 Kronecker delta, 7 poset, 159 Nijenhuis, Albert Hook Formula, 231 Labeled structure, 124 No broken circuit (NBC), 102 disjoint union, 128 Number equivalence, 128 Bell, 10 exponential formula, 132 Catalan, 26 partition, 131 derangement, 43 product, 128 Euler, 120 Sum and Product Rules, 129 Eulerian, 121 Lagrange, Joseph Fibonacci, 2 Wilson’s Congruence, 206 Kostka, 225 Lattice path, 25 Stirling of the first kind (signed), 13 Dyck of semilength 푛, 26 Stirling of the first kind (signless), 12 east step, 26 Stirling of the second kind, 10 endpoints, 25 length, 25 O’Connor, Edmund north step, 26 Wilson’s Congruence, 206 northeast, 26 Odlyzko, Andrew Reflection Principle, 53 Mertens Conjecture, 183 step, 26 Operator Leaf, 23 definite summation, 162 Lemma down, 249 Burnside, 192 forward difference, 162 Deletion-Contraction, 101 up, 249 handshaking, 21 Order isomorphic permutations, 29 Lindström–Gessel–Viennot, 58, 59 Orientation, 103 Length Out-degree, 21 integer partition, 15 lattice path, 25 Part permutation, 4 composition, 16 Lindström, Bernt integer partition, 13 lattice paths, 58 Partition Lindström–Gessel–Viennot Lemma, 58, 59 integer, see also Integer partition Linear recursion with constant coefficients, 97 set, see also Set partition Littlewood, John Pascal’s triangle, 7 Cauchy Identity, 242 Path Log-concave sequence, 55 digraph, 21 Lucas, Édouard graph, 19 congruence, 207 lattice, see also Lattice path Pattern in a permutation, 28 Macdonald, Ian and quasisymmetric functions, 276 symmetric function, 219 Pawlowski, Brendan MacMahon, Percy quasisymmetric pattern avoidance, 276 major index, 76 Permutation, 4 Major index, 76 alternating, 120 Martin, Jeremy ascent, 76 increasing spanning forest, 105 avoid, 28 Matrix-Tree Theorem, 63 coexcedance, 123 Mertens, Franz compatible function, 235 conjecture, 183 copy, 28 function, 183 cycle decomposition, 11 Milne, Stephen cycle of length ℓ, 11 Involution Principle, 49 derangement, 43 Multiset, 8 descent, 75 cardinality, 8 diagram, 29 Index 301

excedance, 122 disjoint union, 145 fixed point, 11 distributive laws, 151 insertion tableau, 242 divisor lattice, 140 inversion, 74 dual, 142 involution, 44 greatest lower bound, 148 Garsia–Milne Principle, 49 Hall’s Theorem, 175 length, 4 Hasse diagram, 140 major index, 76 incidence algebra, 158 matrix, 288 delta function, 159 of a set, 4 Möbius function, 157 one-line notation, 11 reduced, 179, 182 order isomorphic, 29 zeta function, 159, 182 pattern, 28 incomparable elements, 139 recording tableau, 242 isomorphic, 144 representation, 288 isomorphism, 144 reverse compatible function, 270 join, 149 reverse layered, 277 lattice, 149 layer lengths, 277 bond, 167 shuffle, 273 distributive, 151 shuffle compatibility, 274 Fundamental Theorem of Finite standardization, 28 Distributive Lattices, 153 subsequence join irreducible, 152 decreasing, 244 least upper bound, 149 increasing, 244 linear extension, 160 two-line notation, 11 locally finite, 147 Wilf equivalence, 29 lower bound, 148 trivial, 30 lower-order ideal, 143 Petersen, Julius generated by a set, 143 Fermat’s and Wilson’s congruences, 206 maximal element, 142 PIE, 42, 162 maximum element, 142 Pólya, George meet, 148 group action, 200 minimal element, 142 Poset minimum element, 142 푃-partition, 237 Möbius function Fundamental Lemma, 238 and isomorphism, 155 atom, 169 and products, 155 binomial, 178 congruence, 208 Boolean algebra, 140 difference calculus, 162 chain, 139, 147 inclusion and exclusion, 162 maximal, 147 Inversion Theorem, 160 projection, 275 number theory, 163 saturated, 147 one variable, 154 underlying, 275 two variable, 157 characteristic polynomial, 164 operator claw, 169 down, 249 closed interval, 143 up, 249 coatom, 174 order complex, 176 comparable elements, 139 order-preserving map, 144 composition ordinal sum, 146 dominance, 31 partition lattice, 140 lattice, 140 pattern poset, 140 covering relation, 140 product, 146 crosscut, 177 quotient, 169 Crosscut Theorem, 177 homogeneous, 170 definition, 139 Rank Condition, 171 differential, 251 Summation Condition, 171 Dirichlet, 182 support number, 173 302 Index

support set, 173 Hypothesis, 182 ranked, 147 zeta function, 182 corank, 164 Robertson, John rank of a poset, 147 Wilson’s Congruence, 206 rank of an element, 147 Robinson, Gilbert de Beauregard rank set, 147 Hook Formula, 230 reverse 푃-partition, 271 Robinson–Schensted algorithm, 240 Fundamental Lemma, 271 Rooted set, 89 subposet, 142 Rota, Gian-Carlo subspace lattice, 140 binomial poset, 178 total order, 139 Crosscut Theorem, 177 upper bound, 149 twelvefold way, 17 upper-order ideal, 143 generated by a set, 143 Sagan, Bruce Weisner’s Theorem, 176 congruence, 205 Whitney numbers cyclic sieving phenomenon, 209 of the first kind, 156 increasing spanning forest, 105 of the second kind, 156 quasisymmetric pattern avoidance, 276 Young’s lattice, 140 quotient poset, 168 Power series, see also Formal power series representation theory, 287 Prüfer algorithm, 24 shuffle compatibility, 274 Principle Sanyal, Raman Inclusion and Exclusion, 42, 162 combinatorial reciprocity, 107 Reflection, 53 Schensted, Craige Proof increasing-decreasing subsequences, 244 bijective, 6 Robinson–Schensted algorithm, 240 Sedgewick, Robert Quasisymmetric function, 267 analytic combinatorics, 81 algebra, 268 Sequence fundamental, 269 log-concave, 55, 229 monomial, 268 palindromic, 281 pattern avoidance, 276 center, 281 reversal, 280 Redfield, J. Howard unimodal, 54 group action, 200 Set partition, 10 Reflection Principle, 53 block, 10 Reiner, Victor Shareshian, John cyclic sieving phenomenon, 209, 257 chromatic quasisymmetric function, 279 Representation, 287 Signed set, 44 character, 291 Skew partition, 78 defining, 288 Species, 124 dimension, 287 Spencer, Joel equivalent, 288 twelvefold way, 17 inequivalent, 288 Standardization of a permutation, 28 irreducible, 289 Stanley, Richard P. isomorphic, 288 푃-partition, 235 module, 287 (ퟑ + ퟏ)-Free Conjecture, 279 permutation, 288 acyclic orientations, 103 reducible, 289 binomial poset, 178 regular, 290 chromatic symmetric function, 253 submodule, 289 combinatorial reciprocity, 106 nontrivial, 289 differential poset, 248 trivial, 289 quasisymmetric function, 267 trivial, 287 Stanton, Dennis Reverse cyclic sieving phenomenon, 209, 257 plane partition, 233 Statistic, 74 Riemann, Bernhard Eulerian, 121 Index 303

Mahonian, 76 Tiling, 3 Steingrímsson, Einar Transpose Mahonian statistics, 76 Young diagram, 14 Stembridge, John Tree, 22 (ퟑ + ퟏ)-Free Conjecture, 279 Triangle Stirling Pascal, 7 number of the first kind (signed, 13 Stirling of the first kind, 12 number of the first kind (signless), 12 Stirling of the second kind, 10 number of the second kind, 10 Trudi, Nicolò triangle of the first kind, 12 determinants, 227 triangle of the second kind, 10 Twelvefold way, 17 Sum and Product Rules for labeled structures, 129 Unimodal sequence, 54 for sets, 1 Viennot, Xavier for weight-generating functions, 87 lattice path, 58 Symmetric group, 11 Wachs, Michelle Symmetric function, 220 chromatic quasisymmetric function, 279 algebra, 220 topological methods, 176 chromatic, 253 Waring, Edward complete homogeneous, 221 Wilson’s Congruence, 206 elementary, 221 Weak composition, 31 and roots of polynomial, 224 weight-generating function, 86 Fundamental Theorem, 223 Weighting of a set, 86 monomial, 220 Weisner, Louis power sum, 221 Möbius function, 176 Schur, 225 White, Dennis cyclic sieving phenomenon, 209, 257 te Riele, Hermann Wilf, Herbert Mertens Conjecture, 183 equivalence, 29 Theorem generatingfunctionology, 71 푞-Binomial, 78 Hook Formula, 231 Binomial, 72 Word, 5 negative, 88 Cauchy Identity, 242 Young Cauchy–Binet, 63 diagram, 14 Crosscut, 177 cell, 225 Difference Calculus, 162 conjugate or transpose, 14 Euler, 51 outer corner, 241 Fermat’s Little, 206 lattice, 140 Finite Distributive Lattices, 153 tableau Four Color, 100 content, 225 Fundamental Theorem of Symmetric descent set, 271 Functions, 223 Hook Formula, 231 Hall, 175 insertion, 241 Hook Formula, 231 partial, 240 Inclusion and Exclusion, 42, 162 placement, 241 Jacobi–Trudi, 227 Robinson–Schensted algorithm, 241 Maschke, 289 Robinson–Schensted–Knuth algorithm, Matrix-Tree, 63 244 Möbius Inversion, 160 Schensted’s Theorem, 247 Redfield–Pólya, 201 semistandard, 225 Schensted, 247 shape, 224 Weisner, 176 standard, 224 Wilson’s Congruence, 207 Thrall, Robert Zeta function Hook Formula, 230 poset, 159 304 Index

Riemann, 182 Zhuang, Yan shuffle compatibility, 274 Selected Published Titles in This Series

210 Bruce E. Sagan, Combinatorics: The Art of Counting, 2020 207 Dmitry N. Kozlov, Organized Collapse: An Introduction to Discrete Morse Theory, 2020 206 Ben Andrews, Bennett Chow, Christine Guenther, and Mat Langford, Extrinsic Geometric Flows, 2020 205 Mikhail Shubin, Invitation to Partial Differential Equations, 2020 204 Sarah J. Witherspoon, Hochschild Cohomology for Algebras, 2019 203 Dimitris Koukoulopoulos, The Distribution of Prime Numbers, 2019 202 Michael E. Taylor, Introduction to Complex Analysis, 2019 201 Dan A. Lee, Geometric Relativity, 2019 200 Semyon Dyatlov and Maciej Zworski, Mathematical Theory of Scattering Resonances, 2019 199 Weinan E, Tiejun Li, and Eric Vanden-Eijnden, Applied Stochastic Analysis, 2019 198 Robert L. Benedetto, Dynamics in One Non-Archimedean Variable, 2019 197 Walter Craig, A Course on Partial Differential Equations, 2018 196 Martin Stynes and David Stynes, Convection-Diffusion Problems, 2018 195 Matthias Beck and Raman Sanyal, Combinatorial Reciprocity Theorems, 2018 194 Seth Sullivant, Algebraic Statistics, 2018 193 Martin Lorenz, A Tour of Representation Theory, 2018 192 Tai-Peng Tsai, Lectures on Navier-Stokes Equations, 2018 191 Theo Bühler and Dietmar A. Salamon, Functional Analysis, 2018 190 Xiang-dong Hou, Lectures on Finite Fields, 2018 189 I. Martin Isaacs, Characters of Solvable Groups, 2018 188 Steven Dale Cutkosky, Introduction to Algebraic Geometry, 2018 187 John Douglas Moore, Introduction to Global Analysis, 2017 186 Bjorn Poonen, Rational Points on Varieties, 2017 185 Douglas J. LaFountain and William W. Menasco, Braid Foliations in Low-Dimensional Topology, 2017 184 Harm Derksen and Jerzy Weyman, An Introduction to Quiver Representations, 2017 183 Timothy J. Ford, Separable Algebras, 2017 182 Guido Schneider and Hannes Uecker, Nonlinear PDEs, 2017 181 Giovanni Leoni, A First Course in Sobolev Spaces, Second Edition, 2017 180 Joseph J. Rotman, Advanced Modern Algebra: Third Edition, Part 2, 2017 179 Henri Cohen and Fredrik Strömberg, Modular Forms, 2017 178 Jeanne N. Clelland, From Frenet to Cartan: The Method of Moving Frames, 2017 177 Jacques Sauloy, Differential Galois Theory through Riemann-Hilbert Correspondence, 2016 176 Adam Clay and Dale Rolfsen, Ordered Groups and Topology, 2016 175 Thomas A. Ivey and Joseph M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition, 2016 174 Alexander Kirillov Jr., Quiver Representations and Quiver Varieties, 2016 173 Lan Wen, Differentiable Dynamical Systems, 2016 172 Jinho Baik, Percy Deift, and Toufic Suidan, Combinatorics and Random Matrix Theory, 2016 171 Qing Han, Nonlinear Elliptic Equations of the Second Order, 2016 170 Donald Yau, Colored Operads, 2016

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/gsmseries/. This book is a gentle introduction to the enumerative part of combinatorics suitable for study at the advanced undergraduate or beginning graduate level. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature which has never before appeared in print, such as the use of quotient posets to study the Möbius function and characteristic polynomial of a partially ordered set, or the connection between quasisymmetric functions and pattern avoidance. The book assumes minimal background, and a first course in abstract algebra should suffice. The exposition is very reader friendly: keeping a moderate pace, using lots of examples, emphasizing recurring themes, and frankly expressing the delight the author takes in mathematics in general and combinatorics in particular.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-210

GSM/210 www.ams.org